# Introduction To Complex Numbers

A complex number is an integral part of the number system. Mathematically, a complex number is the cumulation of real and imaginary numbers.

A complex number comprises two parts; a real part and an imaginary part. Complex numbers have vast applications, such as electrical engineering and quantum mechanics.

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## Notation Of Complex Number

z = a + bi

This is the notational representation of a complex number z.

Where;

• z is a complex number
• a is the real part of the complex number
• b is the imaginary element of the complex number
• i is the Greek alphabet that denotes the formation of an imaginary part
• The value of i is √-1

Considering this notation, one can easily differentiate the parts of a complex number. Moreover, the set of the complex number is denoted by C.

Let us take an example to demonstrate the complex number.

3 – 7i

In this complex number system, 3 is the real part, and -7 is the imaginary part.

Remember that you do not have to write -7i to demonstrate an imaginary part of a complex number system. However, the number with the i symbol depicts the imaginary part.

## Understanding iota: Imaginary Number

iota (i) is a Greek alphabet primarily used to denote the imaginary part of a complex number. The value of i is √-1. Therefore, i2 is equal to -1.

This interpretation of i is used to solve polynomial equations involving complex numbers. Iota is the major element that helps differentiate a complex number’s real and imaginary parts.

## Why Do We Need Complex Numbers?

The system of complex numbers is essential in mathematics because it is an excellent way to express wave functions without breaking mathematical rules.

Complex numbers and complex functions have been expanded into a rich theory called complex analysis. They have become a powerful tool for answering many tough questions in mathematics, theoretical physics, and many areas (Source: Carleton University)

We need complex numbers because real numbers generally fail to solve certain mathematical problems. Furthermore, the scope of the real number is also limited in practical mathematical calculations.

In the 17th century, mathematician Rene Descartes coined the terminology of complex numbers. The complex numbers were widely accepted by the 18th century. The following example will let you understand why complex numbers are required (Source: The Stanford Encyclopedia of Philosophy)

x2 = -1

Consider the above-mentioned equation. Now, you cannot find the roots of this equation using real numbers. It is because the square of real numbers will never render a negative value. Therefore, complex numbers come into play to solve these types of polynomial equations that are out of the scope of real numbers. The roots of this polynomial equation are complex numbers.

## Properties Of Complex Numbers

• If the sum and multiplication of two complex numbers result in a real number, then the two complex numbers are conjugate of each other.
• Furthermore, the addition of conjugate complex numbers is a real number.
• The multiplication of conjugate complex numbers is also a real number.
• The complex numbers obey the commutative law and associative law of addition and multiplication.
• The complex numbers also apply perfectly to the distributive law.

z1 + z2 = z2 + z1

#### Commutative Law of Multiplication

z1 × z2 = z2 × z1

(z1 + z2) + z3 = z1 + (z2+z3)

#### Associative Law of Multiplication

(z1 × z2) × z= z1 × (z2 × z3)

#### Distributive Law

z1 × (z2+z3) = z1 × z+ z1 × z3

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